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Archimedean semigroup (Definition)

Let $ S$ be a commutative semigroup. We say an element $ x$ divides an element $ y$, written $ x \mid y$, if there is an element $ z$ such that $ xz = y$.

An Archimedean semigroup $ S$ is a commutative semigroup with the property that for all $ x, y \in S$ there is a natural number $ n$ such that $ x \mid y^n$.

This is related to the Archimedean property of positive real numbers $ \mathbb{R}^+$: if $ x, y > 0$ then there is a natural number $ n$ such that $ x < ny$. Except that the notation is additive rather than multiplicative, this is the same as saying that $ (\mathbb{R}^+, +)$ is an Archimedean semigroup.



"Archimedean semigroup" is owned by mclase.
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See Also: Archimedean property

Also defines:  divides, Archimedean
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Cross-references: multiplicative, additive, real numbers, positive, Archimedean property, natural number, property, commutative semigroup
There are 70 references to this entry.

This is version 1 of Archimedean semigroup, born on 2002-11-05.
Object id is 3572, canonical name is ArchimedeanSemigroup.
Accessed 11850 times total.

Classification:
AMS MSC: 20M14 (Group theory and generalizations :: Semigroups :: Commutative semigroups)
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